# Find units of Z[w]

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• Jan 27th 2008, 07:00 AM
tttcomrader
Find units of Z[w]
Let $\displaystyle w = \frac {-1}{2} + \frac {3^{1/2}}{2}i$, find units of Z[w].

1 is the unity, so it is a unit. And what else...?
• Jan 27th 2008, 08:11 AM
ThePerfectHacker
Quote:

Originally Posted by tttcomrader
Let $\displaystyle w = \frac {-1}{2} + \frac {3^{1/2}}{2}i$, find units of Z[w].

1 is the unity, so it is a unit. And what else...?

The the previous excercises $\displaystyle a+bw$ is a unit if and only if $\displaystyle N(a+bw) = a^2+b^2 - ab = 1$. Thus, $\displaystyle a^2 - ab + (b^2 - 1) = 0$ we require that the discrimant, $\displaystyle b^2 - 4(b^2 - 4) > 0$, thus, $\displaystyle 3b^2 < 4 \implies b=0,1,-1$, now you can solve for $\displaystyle a$.
• Jan 27th 2008, 09:23 AM
tttcomrader
I don't understand how do you get to $\displaystyle b^2 - 4(b^2 - 4) > 0$, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

Thanks.
• Jan 27th 2008, 12:59 PM
ThePerfectHacker
Quote:

Originally Posted by tttcomrader
I don't understand how do you get to $\displaystyle b^2 - 4(b^2 - 4) > 0$, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

Thanks.

Yes. I made a mistake it should be 3b^2 < 16. It is just the discrimant of the quadradic.